Norm Inequalities for Derivatives and Differences / Edition 1

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Norm inequalities relating (i) a function and two of its derivatives and (ii) a sequence and two of its differences are studied. Detailed elementary proofs of basic inequalities are given. These are accessible to anyone with a background of advanced calculus and a rudimentary knowledge of the Lp and lp spaces.
The classical inequalities associated with the names of
Landau, Hadamard, Hardy and Littlewood, Kolmogorov,
Schoenberg and Caravetta, etc., are discussed, as well as their discrete analogues and weighted versions. Best constants and the existence and nature of extremals are studied and many open questions raised. An extensive list of references is provided, including some of the vast Soviet literature on this subject.

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Product Details

  • ISBN-13: 9783540563877
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 3/5/1993
  • Series: Lecture Notes in Mathematics Series, #1536
  • Edition description: 1992
  • Edition number: 1
  • Pages: 152
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.34 (d)

Table of Contents

Introduction 1
1 Unit Weight Functions 3
1.1 The Norms of y and y[superscript (n)] 3
1.2 The Norms of y, y[superscript (k)], and y[superscript (n)] 6
1.3 Inequalities of Product Form 12
1.4 Growth at Infinity 26
2 The Norms of y, y', y'' 35
2.2 The [actual symbol not reproducible] Case 35
2.3 The L[superscript 2] Case 36
2.4 Equivalent Bounded Interval Problems for R 38
2.5 Equivalent Bounded Interval Problems for R[superscript +] 43
2.6 The L[superscript 1] Case 45
2.7 Upper and Lower Bounds for k(p, R) and k(p, R[superscript +]) 47
2.8 Extremals 55
2.9 Continuity as a Function of p 77
2.10 Landau's Inequality for Nondifferentiable Functions 80
3 Weights 84
3.1 Inequalities of the Sum Form 84
3.2 Inequalities of Product Form 95
3.3 Monotone Weight Functions 97
3.4 Positive Weight Functions 104
3.5 Weights with Zeros 109
4 The Difference Operator 117
4.1 The Discrete Product Inequality 117
4.2 The Second Order Case 120
4.3 Extremals 123
References 144
Subject Index 149
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